The Annals of Mathematical Statistics

An Occupation Time Theorem for the Angular Component of Plane Brownian Motion

Simeon M. Berman

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Abstract

Let $Z(t) = (X(t), Y(t)), t \geqq 0$, be the standard plane Brownian motion process. Let $(R(t), \theta_1(t))$ be the polar coordinates of $Z(t)$. Suppose that $R(0) = r > 0$ with probability 1, that is $Z(\cdot)$ starts away from the origin. We shall define the process $\theta(t), t \geqq 0$, as the total algebraic angle traveled on the continuous path $Z(s), 0 \leqq s \leqq t$; we take $\theta(0) = \theta_1(0)$; we have $\theta(\cdot) \equiv \theta_1(\cdot) \mod 2\pi$. Let $f = f(\theta), 0 \leqq \theta \leqq 2\pi$, be a bounded measurable function such that $f(0) = f(2\pi)$. For $r_1 > r$ let $\tau$ be the first passage time of $R(\cdot)$ to the point $r_1$. This note is devoted to the computation of the functional \begin{equation*}\tag{1.1}L(f) = E_{r,\theta}\{ \int^\tau_0 f(\theta_1(t)) dt\},\end{equation*} where $E_{r,\theta}\{\cdot \cdot\}$ is the expectation operator under the condition $R(0) = r, \theta_1(0) = \theta$. This is interpreted as the expected occupation time of a measurable subset of $\lbrack 0, 2\pi\rbrack$ if $f$ is the indicator function of the subset. We find an explicit formula for $L(f)$ as a linear functional on the Hilbert space $L_2\lbrack 0, 2\pi\rbrack$. A preliminary result of interest is presented in Section 2: the random variable $\lbrack \theta(\tau) - \theta(0)\rbrack/|\log (r_1/r)|$ has a Cauchy distribution for any positive numbers $r, r_1$ with $r \neq r_1$. This recalls the independent result of Spitzer that $\lbrack \theta (t) - \theta(0)\rbrack/\frac{1}{2} \log t$ has a limiting Cauchy distribution for $t \rightarrow \infty$ [4]. I thank the referee for his constructive remarks and for the alternate proof of Theorem 2.1 given in Section 6.

Article information

Source
Ann. Math. Statist., Volume 38, Number 1 (1967), 25-31.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177699055

Digital Object Identifier
doi:10.1214/aoms/1177699055

Mathematical Reviews number (MathSciNet)
MR203825

Zentralblatt MATH identifier
0189.51201

JSTOR
links.jstor.org

Citation

Berman, Simeon M. An Occupation Time Theorem for the Angular Component of Plane Brownian Motion. Ann. Math. Statist. 38 (1967), no. 1, 25--31. doi:10.1214/aoms/1177699055. https://projecteuclid.org/euclid.aoms/1177699055


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